Embedding calculus knot invariants are of finite type

Budney, Ryan¹ ; Conant, James² ; Koytcheff, Robin³ ; Sinha, Dev⁴

¹Mathematics and Statistics, University of Victoria, PO Box 1700 STN CSC, Victoria, BC V8W 2Y2, Canada
²Department of Mathematics, University of Tennessee, 227 Ayres Hall, 1403 Circle Dr, Knoxville, TN 37996, United States
³Department of Mathematics and Statistics, University of Massachusetts-Amherst, Leaderless Graduate Research Tower, Amherst, MA 01003, United States
⁴Department of Mathematics, University of Oregon, Eugene, OR 97403, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 3, pp. 1701-1742

Résumé

We show that the map on components from the space of classical long knots to the nth stage of its Goodwillie–Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n−1) knot invariant. We compute the E2–page in total degree zero for the spectral sequence converging to the components of this tower: it consists of ℤ–modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connected sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps and cosimplicial and subcubical diagrams.

Détail

DOI : 10.2140/agt.2017.17.1701

Classification : 55P65, 57M25
Keywords: finite-type knot invariants, calculus of functors, embedding calculus, Taylor tower for the space of knots, configuration spaces, mapping space models, evaluation maps, stacking long knots, cosimplicial spaces, spectral sequences

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